1 More General Transfer Function Models Chapter 6 • Poles and Zeros: • The dynamic behavior of a transfer function model can be characterized by the numerical value of its poles and zeros. • General Representation of ATF: There are two equivalent representations: ( 2900 (4-40) m i i i n i i i b s G s a s = = =

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4 Chapter 6 Note that as expected; hence, the effect of including the single zero does not change the final value nor does it change the number or location of the response modes. But the zero does affect how the response modes (exponential terms) are weighted in the solution, Eq. 6-15. ( 29 y t KM i = A certain amount of mathematical analysis (see Exercises 6.4, 6.5, and 6.6) will show that there are three types of responses involved here: Case a: Case b: Case c: 1 τ τ a 1 0τ τ a < τ0 a <

5 Chapter 6

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7 Chapter 6 1. Zeros • Pole at the origin (1/s term in TF model): results in an “integrating process” Note: Zeros have no effect on system stability. • Zero in RHP: results in an inverse response to a step change in the input • Zero in left half plane: may result in “overshoot” during a step response (see Fig. 6.3). x 1 y 0t inverse response Real axis Imaginary axis

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